I'm thinking not necessarily, but I'm not really sure. Whether a function is differentiable or not depends on whether on not every point on it has a derivative.
So say f is y = x^(1/3), and g is y = -x^(1/3). Both f and g are continuous - the graph of the function is unbroken.
(Technically the definition of continuity is that for every value in the domain the limit from below = limit from above = the value itself but yeah that's not really important.)
f+g is differentiate yeah? f + g just gives y = 0
However, y = x^(1/3) and y = -x^(1/3) are both non differentiable, since at the origin x = 0, the derivative is undefined. You get a divide by zero from memory =P